Q.Explain centre frequency or resting frequency in frequency modulation.

Frequency Modulation: The Intuition

Think about how a siren sounds as an ambulance races past you. The pitch — the frequency you hear — rises as it comes toward you and falls as it moves away. The loudness (amplitude) stays roughly the same; only the rate at which the sound waves hit your ear changes. That is the core idea of frequency modulation: you encode information by wiggling the frequency of a carrier wave, leaving its amplitude untouched.

Now contrast this with AM (amplitude modulation), where you change the height of the wave. AM is like whispering and shouting into a microphone — the loudness varies. FM is like changing the speed of a spinning wheel: the wheel's size (amplitude) stays fixed, but how fast it spins (frequency) goes up and down with the message.

Why would anyone prefer FM over AM? Noise — static, lightning, electrical interference — mostly affects the amplitude of a signal. Since FM keeps amplitude constant, the receiver can ignore amplitude noise entirely. That is why FM radio sounds so much cleaner than AM, and why FM is used for high-fidelity music broadcasts.

The Precise Statement

Here is what each symbol means:

• $A_c$ — the constant carrier amplitude (no amplitude variation)
• $f_c$ — the unmodulated carrier frequency (the "resting" pitch)
• $m(t)$ — the message signal (the information you want to send)
• $k_f$ — the frequency sensitivity constant (how strongly the message pushes the frequency)

The instantaneous frequency of this wave is not $f_c$ anymore. It is:

That is the heart of FM: the instantaneous frequency deviates from the carrier frequency by an amount directly proportional to the message signal. When $m(t)$ is positive, the frequency rises; when $m(t)$ is negative, it falls. The amplitude $A_c$ never changes.

Key Parameters

Frequency deviation $\Delta f$ is the maximum shift away from $f_c$:

For commercial FM radio, $\Delta f$ is typically $\pm 75$ kHz. The carrier itself might be at 100 MHz, but the actual transmitted frequency swings between 99.925 MHz and 100.075 MHz depending on the audio.

Modulation index $\beta$ compares the deviation to the message frequency $f_m$:

When $\beta$ is large (>> 1), the signal is "wideband FM" — it uses a lot of bandwidth but is very noise-resistant. When $\beta$ is small (<< 1), it is "narrowband FM", which behaves somewhat like AM but without the amplitude noise problem.

Why the Integral?

You might wonder why the formula has an integral $\int m(\tau)\,d\tau$ inside the cosine. The reason is that phase is the integral of frequency. If you change frequency, you are also changing the phase angle over time. The cosine function takes phase as its argument, so to express a varying frequency, you must integrate that variation to get the instantaneous phase. That integral is what "accumulates" the frequency changes into a phase shift.

A Simple Example …